After successful completion of the course, students are able to...

• to explain, how Monte-Carlo methods work and which mathematical theories are behind them,
• to use Monte-Carlo methods to solve concrete problems in science, economics and finance.

Monte-Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results of  problems which are difficult or impossible to solve analytically. [wikipedia] 

In this course we focus on the following topics:

• generating random variables
• convergence, error estimates, (law of large number, central limit theorem, concentration inequalities)
• variance reduction
• some basis result about Brownian motion, Itô's integral w.r.t. Brownian motion, Itô's formula,
• basic existence, uniqueness, stability results of stochastic differential equations (SDE), and properties of the solutions
• Feynman-Kac theorems, relation between SDEs and PDEs
• numerics of SDEs: Euler-Maruyama scheme, Milstein scheme, high-order schemes, Multi-level Monte-Carlo
• statistical error in the simulation of SDEs
• simulation of non-linear processes, backward SDEs and non-linear version of Feynman-Kac theorems
• McKean-Vlasov SDE and simulation
• stochastic optimization
• Markov chain Monte-Carlo

• Lecture online via ZOOM (or offline lecture if possible)
• Discussion about exercises.

The course is planned for Master’s students, but advanced Bachelor's students are also welcome. Since the topic of the course is very interdisciplinary and planned for all maths programmes, the lecturer will repeat and summarize the necessary results (from analysis, measure theory, theory of stochastic processes and numerics) at the beginning of the course.

Active participation, exercises.
Possible to improve the grade with an oral exam.

• Necessary: Basic knowledge from the first four semester of the bachelors programmes, especially from measure and probability theory, statistics, as well as differential equations.
• Advantage: Basic knowledge about stochastic processes, stochastic analysis, numerics, and partial differential equations